STRAIN LOCALIZATION IN DUCTILE SINGLE CRYSTALS?esag.harvard.edu/rice/066_AsaroRice_SingleCryst_JMPS77.pdf · material changes or imperfections which lower the strength level or work-hardening

J. Mech. Phys. Solids, 1977, Vol. 25, pp. 309 to 338. Pergamon Press. Printed in Great Britain.

STRAIN LOCALIZATION IN DUCTILE SINGLE CRYSTALS?

By R. J. ASARO and J. R. RICE

Division of Engineering, Brown University, Providence, R.I. 02912, U.S.A.

(Received 10th May 1977)

SUMMARY

THIS PAPER is concerned with an analysis of strain localization in ductile crystals deforming by single slip. The plastic flow is modelled as rate-insensitive, and localization, viewed as a bifurcation from a homogeneous deformation mode to one which is concentrated in a narrow ‘shear band’, is found to be. possible only when the plastic hardening modulus for the slip system has fallen to a certain critical value h,,, sensitive to the precise form of the constitutive law governing incremental shear. We develop the general form of this constitutive law, incorporating within it the possibility of deviations from the Schmid rule of a critical resolved shear stress, and we show that h,, may in fact be. positive when there are deviations from the Schmid rule. It is suggested that micromechanical processes such as ‘cross-slip’ in crystals provide specific cases for which stresses other than the Schmid stress may influence plastic response and, further, there is an experimental association of localization with the onset of large amounts of cross-slip. Thus, we give the specific form of h,, for a constitutive model that corresponds to non-S&mid effects in cross-slip, and we develop a dislocation model of the process from which we estimate the magnitude of the parameters involved. The work supports the notion that localization can occur with positive strain-hardening, h,, > 0, and the often invoked notions of the attainment of an ideally-plastic or strain-softening state for localization may be unnecessary.

1. INTRODUCTION

THERE ARE NUMEROUS examples in which plastic flow in crystals gives way from a more or less homogeneous mode of deformation to one that is heavily concentrated in a narrow shear band. This localization may occur either before or after the onset of necking in a tension test, and is often a direct precursor to ductile rupture through profuse void formation and growth within the band. Indeed, in crystals which do not cleave, it generally seems that fracture occurs either by necking down to a ‘chisel edge’ separation or by rupture within such concentrated shear zones. In this paper, we explore the precise conditions that allow an assumed homogeneous pattern of plastic flow, in a rate-insensitive crystal undergoing single slip, to develop bands of localized plastic deformation.

Our results are given in Section 3 and take the form of expressions for the critical value h,, of the plastic strain-hardening modulus h, in that localization occurs when h has fallen in magnitude to h,,. As revealed in a preliminary analysis of the problem by RICE (1977), h,, is dependent on constitutive parameters which correspond to deviations from the Schmid rule of a critical resolved shear stress, as a criterion for continued plastic shearing. Deviations from the Schmid rule are specific examples

t This work was supported by the U.S. Energy Research and Development Agency under Contract EY-76-S-02-3084.

21 309

310 R. J. ASARO and J. R. RICE

of deviations from the ‘normality flow rule’ of continuum plasticity, and it has recently been shown (RUDNICKI and RICE, 1975, and RICE, 1977) that such deviations from normality allow the possibility of a positive, versus an essentially zero (i.e. non-hardening state) or negative (i.e. strain-softening state), value of h,, at localization.

In Section 1.1 we discuss some perspectives on localized flow which explain our approach in the light of existing empiricism and analyses, and follow, in Section 2, with a precise formulation of the incremental constitutive law for single slip. This enables us to calculate h,, in Section 3, and in Section 4 we introduce a dislocation model for non-Schmid effects at the onset of cross-slip in f.c.c. crystals, as a means of estimating some of the continuum parameters on which h,, is found to depend.

1.1 Perspectives on localized flo w

Discussions of localized plastic deformation in the experimental literature have generally associated shear localizations with traction maxima, whether brought on by strain-softening (e.g. ARGON, 1973, and HORNBOGEN and GAHR, 1975), adiabatic heating effects (CHIN, HOSFORD and BACKOFEN, 1964) or the attainment of ideal or non-hardening plastic states (ARGON, 1973, and LUTJERING and WEISSMAN, 1970). There are many examples which suggest such associations. For example, the shearing of coherent precipitates in age-hardened alloys of aluminum (CALABRESE and LAIRD, 1974), iron (HORNBOGEN and GAHR, 1975), nickel (GELL and LEVERANT, 1968) and titanium (BLACKBURN and WILLIAMS, 1969) seems to be greatly augmented within localized shear bands as illustrated in the micrographs of BLACKBURN and WILLIAMS (1969). Radiation-damaged materials are observed to develop soft ‘channels’ with easy slip caused by the sweeping up of point-defect debris by moving dislocations (e.g. WECHSLER (1973)). Unstable dislocation barriers, which are presumed to break up following an increase in testing temperature, lead to flow localization as reported in the classic work of COTTRELL and STOKES (1955). These examples among many others have been interpreted to suggest that localization is in fact caused by degradations in material strength, in the form of a loss of the material’s work-hardening capacity.

In related studies, JACKSON and BASINSKI (1967) uncovered some interesting examples of localized flow in pure copper tested for latent hardening. When the crystals were sheared on slip planes other than the primary plane of prestraining, the initial deformation was concentrated in intense shear bands which occasionally propagated as Liiders bands. Although these subsequently slipped planes were only rarely observed to strain-soften, they were characterized by low work-hardening rates. HOWE and ELBAUM (1961) tested aluminum crystals at elevated temperatures where the work-hardening rates often approached zero and observed that the flow was decidedly non-uniform, consisting of coarse shear bands. The fact that these shear bands are, in many cases, persistent lends support to their association with material changes or imperfections which lower the strength level or work-hardening capacity. But whether or not persistence is actually a result of ensuing substructural changes in the post-localized state, rather than a manifestation of pre-existing imperfections, is usually unknown. Indeed, there are important examples for which explanations based on an approach to ideal plasticity or strain-softening do not seem appropriate.

Strain localization in ductile single crystals 311

For example, ELAM (1925) and later KARNOPP and SACH~ (1928) found that age-hardened alloys of aluminum displayed pronounced shear instabilities following uniform deformation. BEEVER.V and HONEYCOMBE (1962), followed by PRICE and KELLY (1964), made careful studies of this phenomenon and found that the bands, which sometimes formed within diffuse necks, lead directly to a ductile shear type fracture. But Price and Kelly observed that these localizations occurred with increasing load (positive work-hardening), identically in tension and compression, and thus without attendant ideal plastic or non-work-hardening states. Furthermore, they pointed out that the bands, just after forming, were notpersistent, although continued straining would induce rupture within them. Indeed, unloading and polishing away the slip steps followed by reloading caused subsequent localization, but ‘. . . the bands never occupied the same place as before polishing . . .’ (PRICE and KELLY, 1964, p. 986).

The situation in pure face-centered cubic crystals may be similar. SAIMOTO, HOSFORD and BACKOFEN (1965) found that crystals of pure copper fractured, after some diffuse necking, along shear bands whose pattern closely resembled the ideally- plastic slip-line field given by ONAT and PRAGER (1954) with its velocity and, hence, plastic-strain discontinuity. However, as shown by hardness probes, the shear zones contained material that continuously hardened and thus, in this case, there did not seem to exist an ideally-plastic state.

In view of the above experimental observations, it seems prudent to examine the possibility that localized plastic deformation may, in some circumstances, arise for reasons other than work-softening or related local degradations in strength. Speci- fically, we explore, as stated previously, the possibility that localization is the result of a constitutive instability and is predictable from the pre-localized constitutive law relating stress increments to strain increments. The basic mechanics of this approach to localization was developed in the context of finite elasticity by HADAMARD (1903) and extended to elastic-plastic solids, modelled as rate-insensitive, by THOMAS (1961) HILL (1962), and MANDEL (1966). Further, the specific calculations by RUDNICKI and RICE (1975) for frictional solids, and by RICE (1977) for a wide class of ductile material models, including single crystals, reveal that localizations can occur with positive hardening when there are deviations from the normality flow-rule.

In the present context, it is well to note that experimental observations link the onset of localization to the ease of cross-slip or similar micromechanical processes that allow microscopic obstacles to slip to be overcome. This association with cross-slip was suggested by PRICE and KELLY (1964) in their work on aluminum alloy crystals, and profuse cross-slip has long been associated with the onset of the so-called stage III deformation, during which coarse shear bands develop in pure single crystals (CAHN, 1951). As RICE (1977) noted, the stress-state dependence of conditions for the onset of cross-slip should, in general, entail deviations from the Schmid rule of critical resolved shear stress, and hence from the normality flow-rule. In this sense there is a strong theoretical basis for a wider exploration of cross-slip and similar processes as a basis for localization, via the destabilizing effect of non- normality. We explore this connection here.

Our materials are considered to be explicitly rate-insensitive with piecewise linear incremental stress-strain relations. As such, we preclude from the present study not only strain-rate effects but also the possibility that new physical mechanisms


of deformation set-in abruptly and degrade the strength. Clearly, there are cases where unstable flow is influenced by imperfections or initial non-uniformities of material properties, but these are also omitted from present considerations. The localization criteria are worked out for an assumed class of materials that essentially obey Schmid’s rule but display modest departures from it.

Standard notations are used throughout. Bold-face symbols denote tensors or dyads, the order of which is indicated by the context. The magnitude of a vector, such as b, or any of its components, such as b,, is denoted as b or b, and similarly for the elements of higher order tensors. The summation convention is used and commas imply differentiation with respect to the corresponding spatial variable, e.g. tij, i E at*j/axi. For brevity we use dots and double dots to indicate the following products :

P'll = Pijnj, Q:a=Qjiaij; also,

LZ : P = 9ijkiPlk, a’P = aijPjk.

2. CONSTITUTIVE RELATION FOR A CRYSTAL UNDERGOING SINGLE SLIP

The constitutive relation governing an increment of deformation is formulated in accord with the general analysis of crystalline slip by HILL and RICE (1972). With reference to Fig. 1, in the increment of time dt an element of the crystal deforms from the configuration at the lower left of the Figure to that at the upper right. The infinitesimal strain and rotation of the element are D dt and n dt, respectively, where D is the symmetric rate of stretching tensor and a the antisymmetric spin tensor. We observe that the corresponding change of a line element 6x that connects material points of the crystal is

d(6x) = (D dt+Q dt).bx, (2.1)

and, further, that in terms of gradients of the velocity vector v,

20, = vi, j + Vj, i, 2Rij = 2)i, j- Uj, I. (2.2)

Now, as illustrated in Fig. 1, the strain D dt and rotation n dt can be realized

total deformation

attice deformation

I I+ i?! plastic -r-d shear

I r-t =

FIG. 1. The total deformation D dt of a crystalline material in the time interval dt can be kinematically decomposed as follows: (i) A plastic shear dy, imposed on the material under conditions for which the lattice is rigidly fixed relative to the background reference axes, (ii) A lutrice deformation D*dt, imposed under conditions for which the material and lattice deform and rotate identically. The total intinitesi- ma1 rotation of the material is n dt, but the corresponding lattice rotation is Ck*dt.


by the following sequence of operations. (i) The material is given a plastic shear relative to the lattice of amount dr under conditions for which lattice orientations and spacings are held fixed. This shear takes place on a family of crystal planes having the unit normal m, and has the direction of a unit vector d lying in one of the slip planes, so that the alteration of the material line element 6x is

d(6x) = (dy&Vz) * 6x. (2.3)

(ii) Next, the lattice and material deform as one with the infinitesimal strain D* dt and rotation a* dt, accomplishing the further alteration

d(6x) = (D* dt+lR* dt).bx (2.4)

of the line element. We refer to D* and Q* as the rates of lattice stretching and spin, respectively. In general, they are non-compatible and hence not derivable from a velocity field.

Observing that deformation (2.1) is equivalent to the sum of (2.3) and (2.4), and defining j as dy/dt, we have TAYLOR’S (1938) relations

D = D*+Pj, where 2P = 4~ + KRKS,

n = Q*+Wj, where 2W = dm--md. > (2.5)

We assume that elastic response properties, phrased relative to directions embedded in the lattice, are unaffected by slip. Then, following HILL and RICE (1972), the stress rate is related to D* by an expression of the kind

g*+o tr (D*) = A?: D*. (2.6)

Here, Q is the Cauchy (or ‘true’) stress tensor whereas g* is its corotational, or Jaumann, rate formed on axes which spin with the lattice. Specifically,

;* = ir-&&*.o+o.n* , (2.7)

where 6 is the ordinary time-rate of (r following the material element, and the Cartesian components of G* are the ordinary time-rates of the components of o on axes that rotate rigidly at the lattice spin rate Q *. The trace operator tr (D*) gives D/$. The tensor L& (= ~~jkl) of incremental elastic moduli is symmetric in the indices i, j and k, I and, further, has the symmetry

sijkl = 91clij (2.8)

when a strain energy function exists for elastic response. We note in passing that the left side of (2.6) is the lattice-corotational rate of

Kirchhoff stress, defined as ap,/p (where p is mass density and p. is its value in some reference state), when the reference state is chosen to coincide instantaneously with the current state.

Expressing D*, Q* in terms of D, Sz and p by (2.5), the constitutive relation (2.6) takes the form

:+a tr (D) = S:(D-P’j), (2.9) where

P’ = P+9-‘:(W*a-a.W). (2.10)

In this, g is the corotational stress rate formed on axes spinning with the material


(i.e. it is defined by the right side of (2.7) with SL replacing a*), and 9-r is the inverse of A? and has the same symmetries in its indices as does 55’ itself.

The distinction between P’ and P, pointed out earlier by HILL and RICE (1972), is small and involves O(o/_Y)-terms (where cr and S are representative values of stress and elastic modulus) in comparison to O(1). Nevertheless, it is necessary to retain accuracy of the representation to this order if the critical hardening-rate for localization is to be determined to within terms of order cr.

2.1 Plastic shearing with non-Schmid eflects

The constitutive description is completed by specification of an expression for 9. We start with less than the necessary precision in observing that the usual description of the incremental slip relation, in accord with the Schmid concept of a critical resolved shear stress, is in the form

(2.11)

(when the slip system is at yield and subjected to an increment of stress causing further yielding) where dz,, is the increment of resolved shear stress and h is the strain-hardening modulus. This expression for dy must be refined and generalized in the following way. (i) It is necessary to state precisely how dz,, is to be related to a stress increment like g* dt. Different ways of computing dz,, (e.g. as an increment of ‘true’ or of ‘nominal’ or of some other kind of shear stress) lead to differences in the O(o/A?)-terms, in comparison to the O(l)-terms, in this relation (HILL and RICE, 1972). (ii) Next, as remarked in the Introduction, there are good reasons to suspect that small departures from the Schmid description, in the sense that stress increment components other than dz,, affect the shear dy, may be important to the explanation of critical conditions for localization as observed experimentally.

We start with the non-Schmid effects and, letting x be a unit vector perpendicular to d and M in Fig. 1 so that 4, m, x form a right-handed triad, we write in place

of (2.1 l),

dy = ‘h 6% +a,, dr,, +a,,,, dr,, + CI,, dz,, + 2a,, dT,, +2~, dz,,). (2.12)

Here, each CI gives the decrement in the Schmid stress required for flow per unit increase of the corresponding non-Schmid stress, and we will discuss in Section 4 the physical origin of arz and a,,,* in relation to the process of cross-slip. Further, the relation of the d7’s to G* dt that we shall adopt will be given shortly, after considering some of the different possible ways of specifying the Schmid increment dz,,.

In all of these ways we will take z,~ in the form

z md = @l’6’6, (2.13)

where m and 4 are unit vectors at the current instant, but different results for dT,, will arise according to how we choose the vectors & dt and A dt identified in Fig. 1. The most direct manner of choosing these is perhaps to require that 4 and m remain orthogonal unit vectors in the deformation, with 4 remaining in the slip plane as the plane rotates. It is then elementary to show that

i = (D*+ti*).d--(b.D*.ci),

Az = --~.(D*+Q*)+&~.D*.M)~ > (2.14,. 2)


and the Schmid stress-rate C,, (E dr,,/dt) is

2,, = &*Cr~4++?W~~*4+rn~a~~

C&W&* -D*.o+a.D*++z*D**m-d.D*.+J.d. (2~4~)

A second alternative in defining the Schmid rate is to convect the vector d with the lattice-slip plane, so that its length does not remain constant, and to choose m as the ‘reciprocal base vector’ normal to the deformed slip-plane:

2 = (D* +a*) * cs, &z = -m.(D*+Q*). (2.15,. 2)

This gives

L = m$*-D**a+a.D*).d (2.15,)

for the Schmid rate and, following the discussion of an analogous case by HILL and RICE (1972), we observe that dz,, can then be written do: where cry is a mixed component of the tensor cr on convected coordinates which deform with the lattice. A third choice is again to convect CJ with the lattice and to choose m so that it is normal to the deformed slip-plane, but of a length which increases in proportion to slip-plane area. Then,

d = (D* +a*) * 6, & = -m*(D*+a*)+m tr (D*) (2.16~ 2)

and

+a#, = m*[g*+o tr (D*)-D**a+a*D*]*cl. (2.163)

In this case, dz,, is the increment in the product of nominal shear stress on the slip plane and lattice stretch ratio in the slip direction. The case is of special interest because it is this very interpretation of the Schmid stress increment dz,, which RICE (1971) has shown to lead precisely to normality in work-conjugate stress and strain variables. Also, as remarked by HILL and RICE (1972), this expression for draa can alternatively be interpreted as the increment in a mixed component of Kirchhoff stress on convected coordinates, analogous to do: above.

There are unlimited further generalizations of the Schmid stress, and as a fourth and Jinal illustration we again require d and m to be orthogonal unit vectors but simply rotate them rigidly at the lattice spin rate:

;I=W*4, PL = W.&W. (2~7~, *)

In this case,

*,, = ,*;**d. (2.17,)

All the above expressions for the Schmid rate have in common the feature that

fM.3 = m*[g*+o tr(D*)]*d+o:X’:D*, (2.18)

where Z’ is some fourth-rank tensor that depends on the precise way in which the base vectors CJ and m deform with the lattice, and which has components that are O(1). By inverting (2.6), all such rates 2,, can be written in the form

L =(cww+~:~~:LZ’-‘):[&*+otr(D*)]

=(p+a:.%:_F--‘):[g*+atr(D*)], (2.19)

where, in the last version of the expression, we have observed that the first bracketed


term is symmetric and have therefore replaced CLVZ by the plastic flow direction tensor P of (2.5).

We examine now the specification of the various non-Schmid stress increments of (2.12). Each of these increments is multiplied by an a. If the a’s are small fractions of unity, there will be contributions of negligibly small size, of O(olo/Y) in comparison to terms of O(1) or O(a) or O(a/_Y), made to the bracketed term of (2.12) by specifying precisely (e.g. at the level of choosing a specific A?’ in (2.19)) the meaning to be given to the rates of non-Schmid stresses like z,,, z,,, etc. On the other hand, looking ahead to our final result for h,, at localization, if the a’s are not small then the correc- tion due to retention of the @a/_%‘)-terms in the non-Schmid stress rates would in any case be negligible. This is a fortunate circumstance, because our current physical understanding of crystalline slip barely enables a definitive choice of the a’s, much less a precise specification of the non-Schmid stress rates. Indeed, even the precise form of the Schmid rate f,,,, cannot be specified, according to our present understanding of crystalline slip, although in this case we are helped by the remarkable fact that our result for h,,, to the order of accuracy that we determine it, turns out to be independent of how we choose .%‘.

In view of the above remarks it suffices to write expressions of the type

2,s = m-[g*+a tr (0*)1*x, 2,, = d * [g* +B tr (D*)] *d, etc. for the non-Schmid stress increments, since precision of the O(oD*)-terms is unnecessary. Thus, using (2.19), the plastic shear rate of (2.12) can be written

f = $ Q : [G* -t-o tr (D*)], (2.20)

where

Q =P+a:H:944-1+a, (2.21)

and where the tensor a of non-Schmid effects, chosen without loss of generality to be symmetric, has the matrix of components, taken on axes aligned with the triad 4, M, x and ordered in the same sense,

a= EIl %:. sj. (2.22)

The elastic constitutive relation (2.6) and (2.5,) may now be called upon to rewrite (2.20) as

jj =$ Q:LZ’:(D-P$),

and from this we may solve for f as

1

= h+Q:Z:P Q:LZ:D. (2.23)

Finally, by substituting this into (2.9) we obtain the form of the constitutive rate relation needed for the localization analysis,

z+otr(D)= W’:P’)(Q:~) :D

%‘- 1 h+Q:s:P ’ (2.24)


where P’ is given by (2.10), P by (2.5) and Q by (2.21). Also, as suggested above, when the tensor a of non-Schmid effects is chosen as zero and when &’ of (2.18) is chosen to give agreement with (2.16,), one may verify that Q = P’ so that the bracketed tensor of constitutive moduli in (2.24) is then said to exhibit ‘normality’.

3. CONDITIONS FOR LOCALIZATION

HILL (1962) has presented the general theory of bifurcation of a homogeneous elastic-plastic flow field into a band of localized deformation (or, in HADAMARD’S (1903) terminology, into a ‘stationary discontinuity’). There is first the kinematical restriction that for localization in a thin planar band of unit normal n (see Fig. 2)

FIG. 2. Cartesian coordinates aligned with unit vectors of crystal slip system. Surface of localization is shown with unit normal n, having components nI, na, n3.

the velocity gradient field or,, inside the band can differ from that outside, namely 0 vi, j, only by an expression of the form

V~,j-V~ j = giilj.

In addition, there is the requirement of continuing equilibrium that

(3.1)

. -0 WJij-niaij =o (3.2)

at incipient localization, where i is the stress rate within the band and 6’ that outside it. (HILL (1962) actually writes equation (3.2) in terms of a nominal stress rate but, as remarked by RUDNICKI and RICE (1975), the above form is equivalent, given the kinematic restriction which must apply simultaneously.)

If the constitutive rate relation is imagined to have the form

&‘rj = Cijkt vk, b (3.3)

and if the same set of constitutive coefficients C apply both inside and outside the band at incipient localization, then (3.1) and (3.2) will be satisfied simultaneously if

tni cijkl %bk = O. (3.4)

Thus, the critical condition for localization on a plane of normal n is first met when

det (n-C-n) = 0, (3.5)

where n.C*n is considered to be a second rank matrix. Of course, once the critical value of some constitutive parameter, say h, for localization is known as a function of n from (3.5), it is then necessary to determine the orientation n at which the critical state is first achieved.


To carry the localization calculation out for our single-slip constitutive relation of (2.24), we first identify C by rewriting the relation

ir = 9 _ W:P')(Q:-V h+Q:LZ:P 1 :D+C2~o-a~S2-a tr (D) (3.6)

and recognizing that D and n can be expressed in terms of the velocity gradient. Indeed, to form the expression analogous to (3.4) we multiply (3.6) from the left with n* and write

$0~ +ng) and Z&n--ng)

in place of D and Q respectively, to obtain

0 = (n.9.n) - (n*~:P’)(Q:9-4 ag+Aeg h-t-Q:S:P 1 , (3.7)

where

A = +[(n*o*n)I-a-(n*cr)n-n(a*n)]

and where I is the second rank unit tensor, (I)ij = aij.

(3.8)

Rather than attempt to set directly the determinant of coefficients to zero in (3.7) we follow a general procedure outlined by RICE (1977, pp. 214-215) which leads more directly to a solution. We let (n*Z*n)-’ denote the second rank tensor having elements which are the matrix inverse of those of (n.2.n); this inverse may be assumed to exist since, for all cases of present interest, the elastic response properties are remote from any critical values for localization. Multiplying (3.7) by the inverse, we obtain

o = CI+(n.y;4nn)-l .A, _ C(n.~.n>-'.(n.~:P')l(Q:~.nn) h+Q:Z:P

.g. (3 g)

Now, since A of (3.8) has elements which are of order 0, the bracketed tensor

[I+(n*Lt?*n)-’ *A]

differs from the unit tensor only by O(o/8)-terms which, in representative cases, are minute fractions of unity. Thus, we may assume that this tensor has an inverse and calculate it to any desired degree of accuracy by the series

~I+(n~Lt’~n)-l~A]-’ = I-(n~.S’~n)-l*A+ [(n~S’*n)-’ *A]*[(n~LZ~n)-‘*A]-. . . . (3.10)

Thus, when (3.9) is multiplied by this inverse, we have an expression in the form

I- ab

h+Q:L&‘:P 1 *g=o, (3.11)

where the vectors a and b are given by

a = [I+(n*S?*n)-’ *A]-’ *(n*S’.n)-’ *(n*Sf’: P’), b = Q:Z*n. (3.12)

Upon multiplying (3.11) on the left by b* , we obtain

b*a I-__~ 1 h+Q:S’:P (bag) = 0. (3.13)


Now, in view of (2.23), the term beg cannot vanish for non-zero g unless the bifurcation mode g involves no plastic strain. Thus, the only relevant condition allowing a non-zero g is that the coefficient of b-g vanish in (3.13), which gives the critical h at localization as

h+Q:Y:P = b*a. (3.14)

It is easy to verify from (3.11) that the corresponding bifurcation mode has the form g cc a.

Thus, when we use the expressions for b and a of (3.12), the critical value of h for localization on a plane of normal n is

h =-Q:~:P+(Q:~~n)~[I+(n~~~n)~‘~A]~’~(o~~~n)~’~(n~~:P’). (3.15)

3.1 Expansion to order of TV

To review the origin of the terms appearing in (3.15), P is O(1) and is the plastic flow direction tensor of (2.5). P’ is defined by (2.10) and differs from P by terms of 0(0/Y) which involve the plastic spin direction tensor W of (2.5). A is defined by (3.8) and is O(a). Q is defined by (2.21) and involves the term P, of O(l), a term of 0(0/B) involving the fourth rank tensor .%? introduced in (2.18) to account for lattice deformation effects on the Schmid stress-rate, and the term a of (2.12) and (2.22) which accounts for non-Schmid stress effects on yielding. We recall that our specification of Q neglected terms of O(aa/9), which, in any event, one would be hard-pressed to specify. For this reason we should delete from the right side of (3.15) all terms of the orders

ag, aa2/2’, ao3[6R2, . . . , and 02/_Y, a3/LZ2, . . . .

When this is done, with the help of (3.10) and the expressions for P’ and Q just mentioned, there results

k = {-P:Z:P+(P:s*n).(n*LZ*n)-‘*(n*Y:P))+

{-a:Z:P+(a:Y*n)*(n*~*n)-‘*(n.LZ:P)}+

(-o:#:P+(a:X*n)*(n.ZZ*n)-’ *(n.LZ:P)+

(P:~.n).(n.~.n)-‘.[n.(W.a-a.W)]-

(P:~.n).(o.~.n)-‘.A.(n.~.n)-‘.(n.~:P)), (3.16)

and in this arrangement the first set of brackets contains terms of O(Y), the second those of O(a9) and the third those of O(C). Indeed, it is convenient to represent these bracketed expressions as the three terms on the right side of

h = LZF,(n) +a_YeF,(n) +uF2(n). (3.17)

Here, the functions F are all O(1) and 9, a and c are representative members of the corresponding tensors.

To calculate the most critical orientation n, we begin with consideration of the case for which both a and C/L?’ are sufficiently small that we approximate (3.16) and (3.17) by

h = YF,(n). (3.18)

320 R. J. Asmo and J. R. RICE

Upon rewriting the first bracketed term of (3.16), we have

64F,(n) = -P:M:P (3.19)

where the fourth rank tensor .Af is defined by

4” = 64-(~.n).(n.~.n)-‘.(n.~), (3.20)

and we observe that A” has the same symmetry of indices as does .5Z’ and, further, that

n*Jlr=N*n=O. (3.21)

In order to draw conclusions from (3.19) we show now that JV is merely the tensor of incremental elastic moduli governing plane stress states in a plane perpendicular to n. Specifically, we let the unit vectors u, v and n form a right-handed triad and we make the definitions

d = 2D,,u+2D,,v+D,,n, (3.22)

(no sum on repeated indices of form u, u or n), and

D’ = D,,uu+D,,(uv+vu)+D,,vv, (3.23)

where Dij(i,j = a, v, n) are components of an arbitrary rate of stretching tensor D on the directions of the corresponding unit vectors. Further, we observe that

D = D’ +&(nd +dn). (3.24)

For an increment of elastic plane stress deformation in the u-v plane, in the sense that the corotational increment of Kirchhoff stress has no components associated with the normal n to that plane, it is evidently necessary that

0 = n-58: D = n-2: D’+(n*Z*n)*d, (3.25)

and hence that

d = -(n*.Y*n)-‘*(n.LZ:D’). (3.26)

Thus the corotational Kirchhoff rate can be written in terms of the stretching rates

D’ in the plane of stressing as

~+otr(D)=?Z:D=_ZZ:D’+(A?~n)*d

= [Z-(LZ’*n)~(n~LY*n)-’ *(n*LZ’)]: D’ = N:D’, (3.27)

where, for ./lr, equation (3.20) has been used. This establishes the interpretation of J(r as the plane-stress elastic moduli tensor and we note that, since lattice distortions are small in all cases that we consider, the quadratic form

I’= D’:.N:D’ (3.28)

may be assumed to be positive definite in D’. On the other hand,

N:D = N:D’+(M-n)*d = M:D’,

by (3.21), and thus the quadratic form V can also be written

v=D:J’-:D. (3.29)


We see, therefore, that V is a positive dcfinitc function of D’ but a positive semi- definite function of D. In particular,

V=D:N:D=O ifandonlyifD’=O. (3.30)

Making application of these results in (3.18) and (3.19), we see that the critical value of h for localization on a plane of normal n must be either negative or zero, the latter occurring when n is chosen so that P, defined in (2.5), has no components in the plane perpendicular to n. It is straightforward to show that there are two, and only two, orientations n which allow this condition of h = 0, and these are given by

case (i) : n = m, and case (ii) : II = 6. (3.31)

Thus, to summarize, when we approximate (3.17) by (3.18) we find that the critical plastic-hardening modulus at the inception of localization is

It,, = 0, (3.32)

and the plane of localization is either the slip plane (i.e. case (i) of n = 89~) or a plane that we shall refer to as the kink plane (case (ii), n = 4).

3.2 Perturbations about slip- and kink-plane orientations

Now, to study the influence of the terms of order a8 and cr in (3.17), we will begin by expanding (3.17) in a series in n, first about n = m and later about n = 4. In carrying out the work, we wish to be mindful of the fact that a/Z is of the order IO-’ or smaller in representative cases, but that the a’s, which we estimate in Section 4 based on a model for cross-slip, may be appreciably larger, perhaps of the order 10-l.

For the perturbation about n = m we write

n=m+z, (3.33)

where E is understood to be small and to be chosen so that n, like m, is a unit vector. Also, we define the second rank tensors

M=C%*LZ*m, E = ~*LZ.E, H = E.~?.E,

and we observe that by the representation of (2.5) for P,

(3.34)

P:_Y*n = d*(M+E), P:cY:P = d.M.4. (3.35)

Further,

(n*9*n)-’ = (M+E+ET+H)-1

= [I+M-‘.(E+ET+H)]-l.M-l, (3.36)

where ET is the transpose of E and we note that the inverse of a matrix product is the product of matrix inverses, taken in inverse order. Since E is small, so also are E and H, and the inverse of the bracketed matrix can be expanded in a series to give

(n.~~n)-i=M-l-M-l.(E+ET).M-l_M-l.H.M-l+

M-‘~(E+ET)~M-l~(E+ET)*M-l+. . ., (3.37)

where the deleted terms are of third and higher powers in E. By using these several results and reading off the various terms of the first set


of brackets in (3.16) comprising al;,(n), we find, after some algebra, that

LEF,(n) = -o.(H-ET.M-’ .E)*d+O(Ya3). (3.38)

Next, by using (3.34) for M, E and H and writing n- m for E, we can write

SF,(n) = -(n-m)~(d~~~d)~(n-m2)+O(de~n-~~3), (3.39)

where the fourth rank tensor ./Z is

&I = ~-(~.~).(rn.~.~m)-‘.(rn.~), (3.40)

and corresponds to the tensor JV of (3.20) when n is set equal to 4%. Since II-+PZ can have no component in the direction of m to the order considered (n and m are unit vectors), we observe from the properties discussed earlier for JV that (3.39) for YF,(n) is a negative definite quadratic form.

Next consider the term adPFl(n) of (3.17), which is defined by the second set of bracketed terms in (3.16). We can write, with (3.20) and (2.5)

crLZF,(n) = -a:N:P = -(a:J(r.rm)*d, (3.41)

and we observe that F,(m) = 0 since .&Y*nz = 0. By expanding the expression for c&?F,(n) with the help of (3.20), (3.33), (3.34) and (3.37), we obtain

aYFl(n) = (a:dl~o)~(n-m)+O(c&~n-~/Z). (3.42)

Finally, OF,(n) of (3.17) is given by the last set of bracketed terms in (3.16). We first calculate Fz(wz), noting that

(&c%%?q’*(m*S:P) = (Wz*~~Wzm)-‘~(m~~~m)~d = d (3.43)

so that the terms comprising the last bracketed set in (3.16) give

aF,(m) =-a:~:[~(mcr+crm)]+(a:~.m).d+d’(m’[f(clm--md)’b- &J*(om--mcl)]}--4. {3[(m.a.m)I-o-(~.a)m-m(a.~~)). 6. (3.44)

Observing that the two terms involving JZ’ cancel one another, since ~ijkl is symmetric in its last two indices, and simplifying the remaining terms, we come to the remarkable conclusion

aF,(m) = 0, (3.45)

which applies irrespective of the several different generalizations of the Schmid stress rate that we have considered (i.e. the result is independent of &?). We observe, therefore, that for any choice of %‘,

aF,(o) = 0<+-ml). (3.46)

By combining (3.39), (3.42) and (3.46), the formula (3.17) for the value of h at localization on a plane having normal n may now be written

h = -(n-m)*(d.d*d)*(n-m)+(a:JX*d)*(n-mm)+ 0(_.Yln--m13, &ln-mj2, +-ml, cm, c?/~), (3.47)

where the order terms include those of (3.39), (3.42) and (3.46) as well as those deleted in writing (3.16) from (3.15). Now, since n-m has no component in the direction of m, to the order considered, and since (with reference to Fig. 2) the


(MM, tim, xm)-components of 4 * A - 4 vanish, because

m*(d’M’d) = (d’M’d)‘rn = 0 (3.48)

by (3.40), we shall henceforth understand the notation d*JZ* d to represent a plane second rank tensor. Specifically, this is a tensor which has only components with the indices 44, CIX, ~5 and XX, and which components agree with those of its three-

dimensional counterpart. The inverse operation to (d*M*o)*q = p is defined only for vectors p and q lying in the 4--x plane of Fig. 2. In this sense, the inverse

(4 * .M * 4)- ' exists as a plane tensor, and

(4.&!*4)*q=p implies q=(4*d*4)-1*p (3.49)

for associated vectors p and q lying in the 4-z plane. In terms of this inverse, the orientation n which maximizes the right-hand side

of (3.47) is

n = ~n+3(4.~.4)-1.(4.~:a)+O(a2, a/9), (3.50)

and when this expression is inserted into (3.47) we find that the critical hardening-rate

at the onset of localization is

h,, = ~(a:~~4)~(4*~*4)-1~(4*~:a)+O(~~, a’/Y, ~2~9). (3.51)

A parallel calculation can be carried out for case (ii), in which we perturb about the kink-plane n = ti. Now the results are given in terms of a tensor

Y= ~-(~.4)'(4'~.4)-l.(cl'~~ (3.52)

The critical orientation is given by

n = 4+~(m~‘4P~~)-1~(m~~:a)+O(~2, u/U) (3.53)

and

h,, = t(a:‘4P.m)~(m.~..)-‘.(m.~:a)+O(~~, d/8, a3_V). (3.54)

Here, (+vz*.Y* m) is understood as a second rank plane tensor with components in the plane of MZ and x, in a sense analogous to that for (4*A*4), above.

It is interesting to note from (3.51) and from the result that ma& = 0, that only the components of a in the slip-plane (specifically, a,,, CI~, = a,,, and a,,) affect the value for h for localization, at least to quadratic order in the Cos. Similarly, from (3.54), only components of a in the kink-plane affect the result. Also, we see that the critical value of h for the onset of localization is indeed positive when the non- Schmid effects, represented by the a’s, are present.

We note that the terms represented explicitly in (3.51) and (3.54) have the order of a2Y. So long as a is much larger than a/Z, the terms of order CCCT and 02/_P represented by O(. . .) in (3.51) and (3.54) will indeed be negligible by comparison to what is retained. On the other hand, the neglected and retained terms are of the same order when the a’s are of the order of a/g. But in this case, h,, will be such a small fraction of a (say, lo-‘a or less) that it is to be expected that local necking, setting in when h is of the order of a, will have long preceded the attainment of conditions for localization. In any event, as we have emphasized in Section 2, it does not seem possible at present to specify the constitutive relation with enough


precision to determine suitably the terms of order c1a and a’/2 that we neglect

in our expressions for h,,.

3.3 Isotropic elastic mod&

Suppose that %’ has the isotropic form

Zrjkr = G(6ik6j,+SaBjk)+A6ij8kl, (3.55)

where G and A are ‘Lame moduli’, G being the shear modulus. Then we note, for example, that

a:LZ-d = 2Ga*d+Ad tr (a), RW.~‘.SW= G(I+mm)+Amrm, sva..JZ’.4= Gcma$hw, 1

where I is the unit tensor, (IJij = a,,, and also that

(RPZ*Z*PPZ)-~= G-‘(I-&c.=z~), where 5 = (A+G)/(A+2G). (3.56)

With such expressions and (3.40) for M we find

da&?‘* cr = G(xx+45~), (3.57)

so that its inverse in the sense discussed earlier is

(cr*&~)-’ = G-‘(xx+r16/45), (3.58)

and

a:Af*cr = 2G{cr,,x+[(25-l)a,,+25a,,]~}. (3.59)

These last two expressions enable us to carry out the perturbation about n = m, in which case we find from (3.50) that the orientation of the plane of localization is

n = ~+a,, x + d C(25 - W,, +%LI~ +O(a’, a/G),

and from (3.51) the critical hardening-rate is

(3.60)

h,, = G 1

& ++C [(25 - i)cc,, +25a,,-J2 1

+ O(aa, a2/G, a3G). (3.61)

In a similar manner, the perturbation about n = d leads, from (3.53) to the

orientation

n = d+a,,x + $ [(25-l)cl,,+25a,,]m+O(a2, a/G)

and, from (3.54), to the critical hardening-rate

(3.62)

h,, = G {

a:, + iF [(2< - l)cr,, +25cr,J2 >

+ O(cta, a’/G, a3G). (3.63)

The results given thus far are for ‘small’ a. It is possible, however, to write out explicitly the entire expressions for YF,,(n) and c&‘F,(n) in the isotropic case. Since such expressions are wanted only when the a’s are large, there is no need to retain


the terms of order d in (3.17) and it suffices to write

h = 8F,(n) +aYF,(n) (3.64)

(note that since a enters linearly in (3.15), there is no truncation in a in this expression). By using (3.55) in (3.16), and noting that

(IL%%I)-’ = G-‘(I-rnn), (3.65)

(3.64) can be written as

h/G=-l+(n* ilpz)‘+(n* d)2-4c(n*eH)2(n* 0)’ + 2 [n*a*(mz + mti)*n - 2<(n * r*n>(n * d)(n * a * n) +(2t - l)(n * m)(n - d) tr (a)].

Further, when we choose the coordinate axes as in Fig. 2 and use nini = becomes

h/G =-(n:+45n:n~)+4n,n25[(1-n:)a,,+(1-nl)cr,,+(1-n:)a,,]- 2n1 n2a,, +2nJn2(l -4M)cr,, +n1(l-4bt)xJ,

(3.66)

1, this

(3.67)

and, in general, the most critical orientation must be accuracy greater than that of (3.60) to (3.63) is required.

3.4 Some particular cases

sought numerically when

Consider first the possibility of pressure sensitivity of the Schmid stress for the onset of plastic flow. Suppose, for example, that

(L,),.set = 70 +KP, (3.68)

where z. is the flow strength at zero pressure, p = -+rkk is the mean pressure and K is a dimensionless parameter. Then, the Schmid stress increment dT,, is replaced by

L, + M%,,, + &,, + &),

and comparing with (2.12),

clii = +cs,. (3.69)

This yields the same h,,-value, either from (3.61) or (3.63)-by taking 5 = 2/3 (corresponding to A = G) as representative, we have

h,, z 0.12u2G. (3.70)

The parameter K is readily interpreted in terms of the difference between yield strengths in uniaxial tension Q, and in uniaxial compression cc. If the slip system is oriented for a maximum resolved shear stress, i.e. at 45’ to the tensile axis so that z IItd = $0, then (3.68) gives

!?& - 30, = MQ, + 3c%L or

0, - 0, = (4K/3)lX% + a. (3.71)

The ‘strength differential’ is 4~/3 times the mean strength. Hence, making the definition SD = 4~13,

h cI z 0*07(SD)‘G. (3.72) 22


Now, localization in crystals is generally observed to take place in the range of hardening between 5 x 10e4 and 5 x 10b3G and if this is to be explained by a pressure sensitivity of yielding it would be necessary to have values of SD in the range of 0.085 to 0.27. Such values seem considerably larger than most observed strength differentials except perhaps in martensitic high-strength steels where SD may approach 0.07 to 0.10 (SPITZIG, SOBER and RICHMOND, 1975). However, in general, we must surmise that it is unlikely that pressure-sensitivity of yielding has a significant effect on the onset of localization in crystals.

Indeed, as we have emphasized earlier, the experimental association of localization with the onset of cross-slip suggests that we examine this and similar cases, in which non-Schmid stresses can aid in ‘triggering’ an increment of plastic flow on the primary slip system, for deviations from normality which may lead to localization. The details of the cross-slip process will be described in Section 4, but, looking to Fig. 3,

FIG. 3. Cross-slip model. Local obstacle is bypassed by partial segments of a screw dislocation on the primary plane constricting and moving a small distance on the cross-slip plane, before continuing primary slippage.

an increment of the shear stress z,, aids the coalescence of the separated partial dislocations comprising a screw dislocation segment, whereas the stress z,, aids the driving of the coalesced screw dislocation segment a small distance along the cross-slip plane, so as to bypass a local obstacle and then continue the shearing of the primary (MZ~) slip system.

We describe these effects by writing

dy = ; [dz,, +a dz,, +/I dz,,], (3.73)

so that tl describes the effect of the stress on the cross-slip plane and /I the effect of the stress tending to coalesce the partial. Comparing to (2.12)

tl dX = U,b = &a, a 9nX = a,, = t-B (3.74)

and other components of u vanish. Thus, the solutions (3.60) and (3.61) for localization in the neighborhood of the slip plane give

n = nz+&ax, (3.75)

and

h cr = ia2G (3.76)


(thus reducing to RICE’S (1976, equation (30)) result) whereas the solutions (3.62) and (3.63) for localization in the neighborhood of the kink-plane give

and

n = 4 +4/L, (3.77)

h,, = +/3’G. (3.78)

Evidently, the largest of the two results for h,, is to be judged as that marking the onset of localization, although the physics of finitely localized deformation would seem to be very different for the slip versus the kink orientation. To fit observations of h,, in the range of 5 x 10m4 to 5 x 10w3G, it is necessary that the largest of a and p lie in the range from approximately 0.04 to O-14. This range of values for hardening rate is chosen to conform to the hardening rates prevailing at the end of stage II deformation in f.c.c. single crystals, h/G % l/300 and early stage III.

Our dislocation model of Section 4 suggests that these ranges do, indeed, encom- pass reasonable values for a and /? when a crystal is deformed to a state at which profuse amounts of cross-slip become possible.

4. ESTIMATE OF NON-SCHMID STRESS PARAMETERS FOR CROSS-SLIP

One approach to the development of detailed constitutive behavior in metals rests on the premise that plastic micro-processes such as cross-slip, particle or forest cutting, etc., are thermally-activated events. For purposes of illustration, Fig. 3 shows a process like cross-slip. As depicted, the critical event requires that extended dislocations first constrict over a finite length and then bow to a critical radius on the cross-slip plane, so that the segment can be driven the necessary bypassing distance by the stress on that plane. Depending on the medium and dislocations involved, one of these two processes might dominate in the calculation of the reversible work AG required to achieve this transition state; relevant material properties would be lattice structure, stacking fault energy, etc. It is reasonable to assume that this reversible work depends upon all the shearing stresses

r nl? T&P z,,, since they all do work in reaching the critical state. However, for the class of processes we have in mind the reversible work is characterized by a strong stress-dependence. For example, for particle cutting in the aluminum-copper alloys discussed earlier, BYRNE, FINE and KELLY (1961) found apparent activation volumes, v (= kT a(ln $/az,,), of the order IO’-IO3 bz where b, is the Burgers vector. In these cases, plastic flow is observed, as is expected from rate theory, to be sensibly rate-insensitive with a correspondingly weak temperature-dependence.

We idealize the behavior as explicitly rate-insensitive and treat AG(z,,, z,,, z,,, p) as a yieldjiinction, in the sense that plastic flow occurs at measurable rate when this function falls to a critical value (essentially zero). This leads to a relation among the arguments of AG of the form (cJ (4.13) and (4.26))

F(L,, 2-3 r,,, P) = 0. (4.1)

Here, p is a structureparameter which increases with straining due to work-hardening (see Fig. 4), and we write

& = p dy, (4.2)


where p may depend not only on p but also on the stress-state under which the shear dy takes place so that (4.2) is, in general, non-integrable. During plastic flow, F must continuously retain its non-zero critical value and thus we have the plastic ‘consistency’ condition

(8F/8z,,) dr,* +(8F/lJz,,) dz, +(W/&,) dz,, = -,u (aF/dp) dr (4.3)

and this has the same form as (3.73) provided we make the identities

h = - p[aF/iYp][aF/az,,] - I, c( = [~F/&,][~F/&,,] - I, (4.4) p = [aF/az,,] [aF/az,,] - I.

dY,A

%

---. P'Pn

z

-a

p=;, \ '\

FIG. 4. The surface AG = 0, in the space of stresses 7mr, 7+ L,&, can be taken as a yield surface. Different yield surfaces correspond to different values of the structure parameter p.

Note that if cross-slip resulted in measurable shear on the cross-slip plane, rather than just that involved in bypassing local obstacles, we would have a dy,, in addition to dy,, (E dy). A relation like dy,, = a dy would be consistent with plastic

normality but this is far too restrictive in general, and so we take dy,, = 0. Certainly, the approximate experimental validity of Schmid’s rule suggests that

the terms tl and /I in (3.73) and (4.4) should be small. However, no estimates are yet available. In Section 4.1 we make approximate estimates of c( and /I based on more- or-less standard techniques in dislocation theory as applied to the model illustrated in Fig. 3. Before passing to this though, we must bring to mind the approximate nature of the linear elastic continuum theory of dislocations when applied to problems that involve interactions among small segments or closely-spaced dislocations. These approximations in the analysis of complex problems are, unfortunately, unavoidable

at present.

4.1 A mechanism for cross-slip

In face-centered cubic crystals the slip systems are of the type { 11 l)(Ja)( 1 IO} (see, for example, HIRTH and LOTHE (1968, p. 288)); a is the lattice parameter. We consider a screw dislocation gliding on the (111) plane with Burgers vector b = (&z)[lOT]. The two partial dislocations form in the sequence

b(l) = (a/6)[2fl] and b(*) = (u/6)[112].

The slip-plane normal is m = (1/,/3)[ 11 I] and the slip direction is 4 = (I /J2)[ 1 Oi] ;


and X, the direction of partial coalescence, is (1/,/6)n2i]. Partial (1) is assumed to be rigidly pinned against a linear barrier and blocks an entire ‘piled up’ array of n such screw dislocations, thus preventing further primary slip. Relaxation takes place by the cross-slip excursion of the lead dislocation onto the cross-slip plane (whose normal is c), allowing the bypass of the obstacle lying in the primary plane. Since movement of partial (1) on the cross-slip plane is energetically prohibited (see HIRTH and LOTHE (1968, p. 735) for an explanation), cross-slip is possible only if the lead dislocation is completely coalesced or if a small dislocation loop with primary Burgers vector is created on the cross-slip plane near the tip of the pile-up. The first possibility has been discussed by several workers (see HIRTH and LOTHE (1968, pp. 733-736) for a bibliography) while the second has been examined by AVERY and BACKOFEN (1963). As it happens, if this secondary dislocation loop is nucleated near the pile-up tip our analysis will model both processes, as we explain shortly.

Partial dislocation (2) moves to coalesce with (1) under the work performing influence of the normal forces T,_,bi2) and z b(‘); b!” = b(‘)- cr = a/2J2 and b(‘) = bc2)- x = a/2J6 (Fig. 3). For the perfez &z)[lOi] dislocation the elastic eiergy can be written as (ASARO and HIRTH, 1973),

E = ~jj bi bj In (R/r,), (4.5)

where X is the energy factor matrix and R and rO are the so-called outer and inner cutoff radii respectively. The cutoff radii are ill-defined within the continuum theory which makes absolute determinations of our energy terms difficult. X, on the other hand, is well defined and for elastic isotropy has the diagonal form

sm+?# = X,, = G/[47r(l -v)] and X,, = G/47c. When the two partials are extended a distance 6, thus forming an included intrinsic stacking fault, the elastic energy is given by (ASARO and HIRTH, 1973)

E = ~, bi”bS” In (R/r,) +&Cij bi2’by’ In (R/r,) +2~, bl”bl2’ III (R/6). (4.6)

Since the stacking fault has an excess free energy I? per unit area the total energy of the extended dislocation is augmented by a term T’(6-r,). When 6 = r,, the dislocation is fully constricted and (4.5) is regained if we interpret the inner cutoff for both partials and constricted dislocation as all equal to rO. Minimizing E by choice of 6 yields

6 = 2E,#, (4.7)

where E,, E Xijbj”by). Next we consider the change in total energy as the dislocation extends and include

the work done by (or against) the ‘applied stresses’,

AE = -2E,, In (6/r,,) +{z,, b!‘)+z_ by))(6- r,,) +J?(6-F,). (4.8)

Minimizing AE by choice of 6 gives

6 = 2E,,/r, (4.9)

i= = r + {T,, b$” +z_ bi2’}. (4.10)

The total reversible work to be done during constriction of unit length would then be

AG = U, 3 2E12 In (2E,,/i?r,,e)+rF,. (4.11)


If the only critical step during cross-slip were this constriction process, then UC would be the AG. Furthermore, since we assume that there exists an equilibrium pile-up of n dislocations driven by z,~, ltll d z b(‘) should be replaced for an isolated single-ended pile-up by nzmd and i= set equal to I- +nz,,bL’) +z,,b!‘). a is now computed from its definition in (4.4) to be (maintaining n constant)

/YI = b;“/(nb;“) = 058/n. (4.12)

The criterion for complete coalescence is obtained by setting AG = 0 in (4.11). Taking T,, M b,, as suggested by HIRTH and LOTHE (1968, p. 762) for f.c.c. metals, and v = 3, the criterion is

n(z,,/G) +0.5 (z,,/G) = 1/8x - 2l-/Gb,. (4.13)

For f.c.c. metals with intrinsic fault energies less than 150 erg/cm’, 2T/(Gb) < 1/8n; for copper, for example, T/Gb z l/250 (HIRTH and LOTHE, 1958, p. 764) and

n(z,,/G) +0*58(z,,/G) z 0.03. (4.14)

Stress ratios of 3 x lo-‘G are never attained in pure metals and are virtually unattain- able with alloying. If n z 10, then z,_, x 3 x 10e3G would in principle be sufficient

to induce coalescence and these stress values are only approached in, say, precipitation-hardened crystals at the onset of profuse cross-slip. Thus, we infer that coalescence is at most possible at the stresses prevailing at the latter stages of deformation but probably only over segments restricted in length and aided by thermal activation. Of course, for metals such as pure aluminum where T/(Gb,) > l/50 we conclude that partial extensions are not probable or are too small to be modelled in our continuum theory. In what follows we assume that UC, which may be driven to nearly vanishing values, is the reversible work required to spread the constriction along the linear barrier as the segment of secondary loop bows out on the cross-slip plane.

Actually the ‘applied stresses’ should not a priori be interpreted as those applied externally. Lattice frictional resistance should be subtracted and if we are considering the motion of dislocations over distances large compared with the spacing of zones in a zone-hardened crystal or the dislocations in a dislocation forest we must also account for their frictional resistance. In the aluminum-alloy crystals discussed

earlier these resistive terms comprise a large fraction of the total flow stress ranging upwards to 4. In writing nz,,b, for the total resolved force on the lead dislocation in the array we have also ignored the interactions among these dislocations and those of nearby glide bands. To include these latter effects we could analyze an infinite sequence shown in Fig. 5(b) as well as the isolated single pile-up shown in Fig. 5(a). Here, p < I and typically experiments suggest that p z O.ll. If both cases are treated

as though the dislocations in the pile-up were continuously spread out as a freely- slipping shear crack we find that the near-tip stress field has the well known form

CzmbY LJ = W7v)-fCcos (S), sin W>19 (4.15)

where the stress intensity factor K is given by

K2 = 2Gnr,,b, for an isolated single-ended pile-up, K2 = $rcGnr,,b, for an isolated double-ended pile-up, K2 = Gnz,,b, for an infinite sequence of double-ended pile-ups.


(a)

(b)

FIG. 5. Arrays of discrete dislocations ‘piled up’ at a barrier at the origin. qJ is the angle of inclination of the cross-slip plane. The dots represent discrete dislocations and the surrounding solid lines the continuous smearing of these dislocations along the slip line for the (a) isolated array and (b) an infinite sequence of arrays, each separated by a distance p; (c) shows an idealization of an array where the lead dislocation is discrete while the remaining dislocations are continuously distributed.

For the double-ended pile-up cases, 12 is interpreted as the number of dislocations of each sign in a length I in a slip line of length 21. For the three cases respectively, n and the slip-line length I are related by n = rcr,,I/(Gb,), n = 2r,,l/(Gb,) and 12 x z,,p/(Gb,) if p/Z is small. In fact, the interactions of the infinite sequence tend to decrease, for a given slip-line length, the near-tip stresses. Now these inter-glide- band interactions are just the sort that form a basis for most analyses of stage II hardening in single crystals (NABARRO, BASINSKI and HOLT, 1964). The force on the

lead dislocation zIb, is given by K2/2G (see RICE (1968, p. 299)) and for the infinite sequence case is equal to nz,,b,/2 which has obvious implications for increased z ,d-values in (4.13) for the coalescence criterion. That is, to meet the condition of (4. I3), z,* for the case of an infinite sequence of pile-ups would have to be a factor (27cZ/p)* times as large as it must be for an isolated single-ended pile-up. The reductions in force on the lead dislocation lead us further toward the view that coalescence, in most cases, is not completed by the stresses but through applied stress, and local fluctuations, occurs only along segments with finite size.

Now, with the above in mind, we assume that there exist locally, along the partially constricted dislocation, minute segments that are ‘pinched’ to total constriction: the energy or reversible work involved can be computed in a fashion similar to that used by STROH (1954). We note that this will yield a work term UP which depends upon the variables and parameters z,~~, r,, and I?. However, as we shall see, this term does not, in our model, contribute to the yield function For to the computations for a or p and hence we will not reproduce the derivation of it here.

The final part of our analysis is concerned with the process of bowing of the constricted (or free) segment to critical conditions on the cross-slip plane. We assume that the dislocation takes on a roughly semi-circular form with radius r. The resolved stresses that do work on the loop are composed of those caused by the pile-up and those due to external sources. From (4.15), this stress becomes (Fig. 5(a))

z,, = K(2np)-* cos (34) +z,, sin 4. (4.16)

Now, as a segment of the lead dislocation begins to move out onto the cross-slip plane, we will be interested in how the stress field of the array is affected. Figure 5(c) illustrates our argument in two dimensions for the case of straight dislocations,


We assume the lead dislocation is discrete while the rest of the array is continuously distributed. The value of the stress intensity Knet at the end of the distributed array is dependent upon the applied stress zmd and the position of the dislocation at <a( = re’“) which, as shown, is able to move on the cross-slip plane with orientation 4. Knet is allowed to relax completely, i.e. K,,,, = 0, and we seek the value for p, say pC, at which a = 0, 4 = 6 and thus for which no further movement of the continuous array is possible. The problem of a dislocation near a single continuous array or slipping line has been considered for elastically isotropic media by RICE and THOMSON

(1974) and by ASARO (1975) for generally anisotropic media. From these results we may write

K,,, = K - Gb,(2nr)-* cos ($0). (4.17)

In order to estimate pC from (4.17) we require an appropriate value for K; to obtain one, we look ahead to our yield function, equation (4.26), where we find that, at yield,

K z &Gbr(er,)-* set (34)

when we leave aside the minor contributions due to the terms in U, and 8. Alterna-

tively, if C#J w trc, then z,/G at yield is O-032. Now, from (4.17), we may write

and, if again 4 w ?&r, then pC M 1_3b,. If 4 = 0, then pC M 56,; and when C$ = 70”,

then z,/G = 0.024 and pC w 2_25b,. In other words, for 4 2 70” (the cases of interest) this ‘collapse’ distance is quite small. We then expect that in our actual

bowing process, so long as the segments bow out to distances comparable to or larger than 2b,, the trailing array will essentially collapse to the blocking line, and we may then use (4.16) to represent the pile-up tip-stress field acting on the cross-slipping dislocation. The interaction of the loop expanding on the cross-slip plane with the pile-up is modelled following an argument due to RICE and THOMPSON (1974). Since a straight dislocation is attracted toward the tip of a slipping line with a force given exactly by .Xijbibj/p (ASARO, 1975), viz. the ‘image force’, we assume that the forces exerted by the pile-up on a segment has an image form. Then the half-loop is acted upon by an image half-loop, and this is taken into account by choosing for the energy of loop expansion the energy per unit length of a full circular loop, namely (HIRTH

and LOTHE, 1968, pp. 144145),

2-v T=Gb,2 ~

87c(1 -v) In (8r/e2ro).

Now, the complete free energy change during the bow-out is given by (taking 8/e2 = l),

U = cDr In (r/ro)-9Y(r3/2-r~/2)+2Ucr-~(r2-r~)+Up, (4.19)

where

@ = Gb,2(2-v)/[8(1 -v)-J, g = 1*4Kb, cos (&$), d = +rcr,, b, sin 4.

The terms containing Z# and d account for the work of the applied stress of (4.16);


and the terms with UC and Up are the energy of the constricted segment. As a and d are increased, the segment traverses configurations, say r = r,(a’, 8) for which the free energy is a minimum-this is diagrammed in Fig. 6. AG is defined in the

U

FIG. 6. Schematic form of the activation barrier for cross-slip.

figure as

AG = U(r,, a’, % U,, g)- U(r,, QD, 9, UC, J), (4.20

where rl is the stable loop radius and r2 corresponds to the barrier. As we desire the derivatives of AG when AG itself vanishes, we have both the conditions

and

when r = r,, the critical radius.

XT/i% = 0 (4.21)

a2Upr2 = 0, (4.22)

Equations (4.21) and (4.22) enable us to solve for the critical value of $8, corresponding to r = rc. Given that Br,/@ (( 1 for representative stress levels, we treat that quantity as a small parameter and solve for r, and W to appropriate orders in the parameter. Thus, defining

5 = er, exp (- 2U,/@) Z er,, (4.23)

the latter approximation being valid for representative values of UC, we find for the critical value of g’,

W = (4@/3rf)(l -Jr,/@) (4.24)

whereas the corresponding critical radius is

r, = r,(l-2&,/Q) Z rl z era. (4.25)

Recognizing from (4.19) that L% and 8 are functions of the stresses, the former via the stress intensity k, equation (4.24) may be regarded as a yield criterion and from it we may compute the non-Schmid factors a and /I. Indeed, to write this yield function it is most convenient to square both sides of (4.24), retaining only the same order of accuracy in the small parameter &‘r,/Q, and replacing the k2 which appears on the left in terms of the stress r1 on the leading dislocation, writing k2 = 2Gz,b,.


In this way we obtain the yield condition

(4.26)

For representative values in f.c.c. crystals, v = 3 and C$ = 70”, and with

rl Z er, z eb,, this becomes

z1 +0*62r,, = 0.024G. (4.27)

Thus, recognizing that z1 is a function of the primary shear stress r,, and parameters describing the geometry of the pile-ups (e.g. the slip-line length r), we compute the non-Schmid parameter a as

CC = 0.62/(1%,/&,,). (4.28)

Hence, pile-up geometries that efficiently concentrate the applied primary shear stress have small non-Schmid effects, and conversely.

For the three pile-up cases discussed earlier we have

71 = (r& UGb,, xr:, I/2Gb,, r:A2Gb,),

respectively. Thus, (4.28), simplified with the help of (4.27), yields

(4.29)

CI = (l*l(b,/l)*, 1*6(b,/l)*, 2.8(b,/p)+) (4.30)

for the three cases of single- and double-ended isolated pile-ups and sequences of closely-spaced pile-ups.

Now, 1 typically ranges from 10e5 to lOA cm in stage II and early stage III hardening and 6, x 2.7 x 10-a cm. With 1 = 10e5 cm we find c( = 0.06 and O-08, respectively, for the model cases of isolated single- and double-ended pile-ups whereas for sequential pile-ups, taking p = O-51, we find a = 0.21. Of course, for I = 10e4 cm, all the numbers are smaller by approximately 31. We note that 2 typically decreases with on-going strain with the implication that the relative

importance of r_, with respect to z,,, and hence the size of tc, increases. Thus, since

h c~ z $a*G, the critical hardening modulus, below which localization occurs, increases with strain in stage II and early stage III hardening, while h itself decreases with the onset of stage III. This pattern continues until the localization condition

is met. For the case of a single-ended pile-up, it could also be appropriate in certain

circumstances to regard the number of dislocations n, rather than 1, fixed in the differentiation of (4.28). Since z1 = nz,,, for the isolated single-ended pile-up, this

leads to

c( = 0.62/n = 2*3(b,/l)“, (4.31)

which is approximately twice the value obtained when I is instead regarded as the fixed quantity (cf. the first member of (4.30)). Alternatively, we may choose n = 10 as representative and this suggests c1 = 0.06 in this case.

To compare the predicted critical hardening-rate ~Gc? with experimental values we need only remember that near the stage II-stage III transition, h/G is of the order l/300 x 3 x 10m3 and, of course, falling with increasing strain. If a % 0.1


on the other hand, which is representative of our estimates, then &?G is of the order 2.5 x 10e3G. Thus, the magnitudes of these non-Schmid effects we have calculated lead to predicted critical hardening-rates that are certainly consistent with experimentally measured rates in the later stages of deformation where localization is observed.

From (4.26) and (4.27) it is easily seen that the yield criterion will not generally require stress levels exceeding those involved in the coalescence criterion, equations (4.13) and (4.14). We have already seen this for 4 x 70” in the discussion immediately following (4.17). Thus, our assumption of incomplete constriction is not violated by our derived yield function. In fact, we may also examine our assumption regarding the magnitude of UC by using the yield criterion and (4.11). For copper, with

rl(Gb,) = l/250, we find that 2U,/@ x 0.04 and thus was justifiably neglected in (4.23). Our critical path, then, is one for which AG vanishes while UC remains positive.

Some final comments on the model are worth making before closing this section. We have seen that the model predicts a vanishing AG with positive UC. This in turn required that some mechanism operate to initiate a local constriction and we sum- marized that thermal fluctuations were important. But, if thermal activation were involved, the model would, of course, predict some degree of rate sensitivity that, in the present case, we wish to preclude. We could avoid this by realizing that the perfectly linear barrier heretofore envisioned in our analysis is simplistic and should be replaced by one having irregularities (e.g. forest dislocations) that enable local constrictions to develop before complete constriction. For the present, we will not include such refinements since there is no clear way in which to model accurately such details and overly speculative attempts to do so would only detract from the main intent of our calculations. The only further observation we make is that since UC is positive both the a and B effects exist together whereas if UC were to vanish, only /3 would remain, unless some extraneous barrier were present, such as the dislocation forest, restricting the free segment length bowing onto the cross-slip plane.

In the case where UC > 0 we may recompute B from (4.26), using (4.23), (4.11) and (4.10) to obtain

B = 0*3&J - ~0)(~12’/~,) set (36) (2alr,)*

or

B = O-3(6 - rJ(~!2’/b,) set (+$).

(pro) + (4.32)

Here, 5 = 1 or ,/2 for the isolated single or double-ended pile-up respectively. If we assume nearly complete constriction (6 x 2r,) and use the same numerology that followed (4.30), we find fi x 0.01 for, as an example, the isolated double-ended pile-up. This can be compared to (4.12) by taking n cz 10 as a typical number to observe that /3, as obtained from (4.32), is substantially lower than the value O-058. Values of B may be computed from (4.1 l)-(4.14) for the other cases we have considered and the general conclusion follows that /? is smaller than CY when both effects coexist.

336 R. J. A~ARO and J. R. RICE

5. DISCUSSION

The particular mode of localized deformation considered here seems to be a ‘limiting’ one in the sense that experimentally it is often preceded by other modes, such as necking. In their experiments on under-aged aluminum-copper alloys, BEEVERS and HONEYCOMBE (1962) clearly demonstrated the two modes of inhomogeneous flow. Necking usually preceded localized shearing but the results of these workers taken together with those of PRICE and KELLY (1964) suggest that it is often localized shearing that occurs first. The two modes seem independent of each other and which one occurs first may well be a matter of specimen shape (see, for example, HILL and HUTCHINSON, 1975). A thorough analysis of the bifurcation modes coupled with a complementary experimental study on this system should be most helpful in developing a fuller understanding of inhomogeneous deformation in general. However, it seems to be the localized shearing that leads to rapid failure of these crystals, and we have demonstrated that this is possible with positive work-hardehing and without work-softening or non-hardening plastic states.

There are several other important aspects of shear localization that require further

understanding. We have explicitly considered the effects of non-normality with a smooth yield surface, in the context of slight deviations from Schmid’s law for yielding. We recall that with a in the range 0.1, h,,/G is predicted to be of the order 2 x 10p3, and this corresponds quite reasonably with what is expected for the hardening rate shortly after the stage II-stage III transition in face-centered cubic crystals (NABARRO et al., 1964). Related studies of localized deformation in polycrystalline sheets (ST~~REN and RICE, 1975) and general solids (RUDNICKI and RICE, 1975) have demonstrated the importance of ‘vertex’ structures on yield surfaces. Single crystal yield-surfaces contain vertex-like structure at points where the yield surfaces for various slip-systems intersect. Indeed, in following the rotations of the tensile axis

during straining, PRICE and KELLY (1964, p. 982) noted that it “. . . usuaZZy reached or ooershot the [OOl]-[ill] boundary . . ., and the slip bands (shear bands) belonged to the conjugate system.” In all cases, the shear bands were closely aligned with the active slip system-a result we reproduce in our analysis. But whether localization is preferred on the conjugate system because of vertex effects in the constitutive law or for the non-Schmid effects explored here, is not as yet clear. Localization is favored by low hardening-rates and, if the latent hardening trends documented for pure face-centered cubic crystals (JACKSON and BASINSKI, 1967) also apply to these zone-hardened crystals, it is expected that the work-hardening on the conjugate planes would be correspondingly low. Again, full resolution of this question awaits

further study. Future experiments on shear fracture in single crystals, and notably precipitation-

hardened single crystals, should prove invaluable for developing more complete analyses of the localization phenomena. Our results suggest that the values of work-

hardening rates when localization occurs as well as stress level should be determined. The experiments to date, that we have mentioned, have noted that the resolved shear stress, but not shear strain, correlates reasonably well with the appearance of localized shearing, however there may well prove to be an equally good corre- lation with h, the slip-plane hardening rate. Furthermore, the persistence of the

shear bands is important in evaluating the role of strain-softening. PRICE and KELLY


(1964) seem to be among the few workers who have posed this question and, for the case of zone-hardened crystals, they found no evidence for strain-softening. However,

there is the possibility that in their aluminum+opper alloys, room-temperature ageing during unloading and repolishing can complicate the interpretations. Evidently, the point is significant and must be explored further.

Finally, we wish to call attention to the symmetry of our predictions regarding the orientation of the plane for localization: both planes that are nearly coincident with the slip plane and with what we have called the kink plane (normal in the slip direction 6) are predicted. It has, of course, occurred to us that a finite localization

of the kink type is kinematically analogous to the phenomenon of ‘deformation kinking’, in which a crystal lattice is observed to bend about an axis along x in a planar band with normal d (CAHN, 1951). The mechanisms for forming such kinks are not well established and have usually been associated with asymmetries in loading. However, our analysis suggests that a rather different explanation may be possible in certain cases, in that some observed kinking may be explainable as a localization instability.

To conclude this paper, we recall its intentions set forth in Section 1. There are numerous examples of highly-localized plastic deformation in ductile single crystals that are not easily explained in terms of strain-softening or ideal plasticity. We have instead put forth a rather detailed and (we hope) sufficiently precise analysis of the problem in initially-homogeneous rate-insensitive crystals, which does seem to provide a suitable description of the process. We have given formulae for the critical parameters, and have discussed in detail how they could be related to the micromechanics of slip. Clearly, our procedures of Section 4 could be applied to other types of ‘triggered’ slip processes. Analytical extensions of our model are appealing, but whether or not our description, or some other, is the correct one in any particular case will probably be determined only by careful experiment of the kind we have suggested.

ARGON, A. S. 1973

ASARO, R. J.

ASARO, R. J. and HIRTH, J. P. AVERY, D. H. and

BACKOFEN, W. A. BEEVERS, C. J. and

HONEYCOMBE, R. W. BLACKBURN, M. J. and

WILLIAMS, J. C. BYRNE, J. G., FINE, M. E.

and KELLY, A. CAHN, R. W. CALABRESE, C. and

LAIRD, C. CHIN, G. Y., HOSFORD, W. F.

and BACKOFEN, W. A.

1975 1973 1963

1962

1969

1961

1951 1974

1964

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